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A226873 Number A(n,k) of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A226873 #23 Sep 06 2018 12:30:42
%S A226873 1,1,0,1,1,0,1,1,1,0,1,1,3,1,0,1,1,3,4,1,0,1,1,3,10,11,1,0,1,1,3,10,
%T A226873 23,16,1,0,1,1,3,10,47,66,42,1,0,1,1,3,10,47,126,222,64,1,0,1,1,3,10,
%U A226873 47,246,522,561,163,1,0,1,1,3,10,47,246,882,1821,1647,256,1,0
%N A226873 Number A(n,k) of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A226873 Alois P. Heinz, <a href="/A226873/b226873.txt">Antidiagonals n = 0..140, flattened</a>
%F A226873 A(n,k) = Sum_{i=0..min(n,k)} A226874(n,i).
%e A226873 A(4,3) = 23: aaaa, aaab, aaba, aabb, aabc, aacb, abaa, abab, abac, abba, abca, acab, acba, baaa, baab, baac, baba, baca, bbaa, bcaa, caab, caba, cbaa.
%e A226873 Square array A(n,k) begins:
%e A226873   1, 1,  1,   1,    1,    1,    1,     1, ...
%e A226873   0, 1,  1,   1,    1,    1,    1,     1, ...
%e A226873   0, 1,  3,   3,    3,    3,    3,     3, ...
%e A226873   0, 1,  4,  10,   10,   10,   10,    10, ...
%e A226873   0, 1, 11,  23,   47,   47,   47,    47, ...
%e A226873   0, 1, 16,  66,  126,  246,  246,   246, ...
%e A226873   0, 1, 42, 222,  522,  882, 1602,  1602, ...
%e A226873   0, 1, 64, 561, 1821, 3921, 6441, 11481, ...
%p A226873 b:= proc(n, i, t) option remember;
%p A226873       `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
%p A226873     end:
%p A226873 A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), n!*b(n, 0, k)):
%p A226873 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A226873 b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n-j, j, t-1]/j!, {j, i, n/t}]]; a[n_, k_] := If[k == 0, If[n == 0, 1, 0], n!*b[n, 0, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, Dec 13 2013, translated from Maple *)
%Y A226873 Columns k=0-10 give: A000007, A000012, A027306, A092255, A092429, A226875, A226876, A226877, A226878, A226879, A226880.
%Y A226873 Main diagonal gives: A005651.
%Y A226873 Cf. A131632, A182172.
%K A226873 nonn,tabl
%O A226873 0,13
%A A226873 _Alois P. Heinz_, Jun 21 2013